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Unformatted text preview: = { x + iy :  x  < / 4 , < y < } , and suppose that  f ( z )  1 and f (0) = 0. Prove that  f ( z )   tan z  for all z . 5. Find a harmonic function in the region { z :  z  < 1 , Im z > } whose boundary values are 1 on the interval (1 , 1) and 0 on the halfcircle. 6. Let f and g be two analytic functions in some region D , and suppose that f ( z ) + g ( z ) is real for all z D . Prove that fg is constant. 7. For all real a , evaluate Z tan( x + ia ) dx. 1 Use the principal value when a = 0. 8. Let f be a meromorphic function in a neighborhood of 0 and an analytic function in a neighborhood of 0 with the properties (0) = 0 and (0) 6 = 0. Prove that res f = res [( f ) ] . 2...
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This document was uploaded on 01/25/2012.
 Spring '09
 Math

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